Why does the 3D view have trouble with clip start <=0.01
From my purely theoretical and limited technical knowledge on the matter real time rendered scenes (like in OpenGL or DirectX) use a Z depth buffer to determine the distance to camera of any object or vertex on screen.

That Z buffer is basically a black and white image of the screen, where for every pixel there can be a a value from 0 to 255 (black to white) for distance information. This image is then used to determine what is visible and what is not and build the viewport image. (There may be other wizardry involved for increased precision like double precision or using other color channels for more possible depth values, by my technical knowledge is limited here)

If the clipping distances (Start and End)are too far apart these 255 (or whatever) shades of grey are not gonna cut it and will not be enough for all the possible depth differences on screen, so artifacts and errors start appearing on screen, like the so called Z-Fighting on co-planar or coinciding faces, or these seemingly ghost-like edges and see-through faces in your mesh.

Now in theory changing clipping values but keeping them in sync should have no ill effects, but there are probably other variables like your graphics card capabilities, drivers, etc.

I would also like to know more about this, so if Blender developer, programmer or someone with more technical knowledge care correct me or enlighten us, I would be very grateful.